where
is a Hermite polynomial (Watson 1933; Erdélyi
1938; Szegö 1975, p. 380). The generating function
(2)
where
is the floor function, can be derived from this
equation (Doetsch 1930; Szegö 1975, p. 380). The more straightforward sum
with
replaced by in the denominator is given by
Almqvist, G. and Zeilberger, D. "The Method of Differentiating Under the Integral Sign." J. Symb. Comput.10, 571-591, 1990.Doetsch,
G. "Integralgleichenschaften der Hermiteschen Polynome." Math. Z.32,
587-599, 1930.Erdélyi, A. "Über eine erzeugende Funktion
von Produkten Hermitescher Polynome." Math. Z.44, 201-211, 1938.Foata,
D. "A Combinatorial Proof of the Mehler Formula." J. Comb. Th. Ser.
A24, 250-259, 1978.Petkovšek, M.; Wilf, H. S.;
and Zeilberger, D. A=B.
Wellesley, MA: A K Peters, pp. 194-195, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.Rainville,
E. D. Special
Functions. New York: Chelsea, p. 198, 1971.Szegö,
G. Orthogonal
Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 380, 1975.Watson,
G. N. "Notes on Generating Functions of Polynomials: (2) Hermite Polynomials."
J. London Math. Soc.8, 194-199, 1933.
![sum_(n=0)^infty(H_n(x)H_n(y))/(n!)(1/2w)^n=(1-w^2)^(-1/2)exp[(2xyw-(x^2+y^2)w^2)/(1-w^2)],](/images/equations/MehlersHermitePolynomialFormula/NumberedEquation1.svg)
is a Hermite polynomial (Watson 1933; Erdélyi
1938; Szegö 1975, p. 380). The generating function
is the floor function, can be derived from this
equation (Doetsch 1930; Szegö 1975, p. 380). The more straightforward sum
with 
replaced by 
in the denominator is given by
